The current article intends to study some elementary constrained approximation aspects of the bivariate fractal functions. To this end, firstly the construction of bivariate fractal interpolation functions available in the literature is revisited with a focus to obtain a parameterized family of fractal functions corresponding to a prescribed bivariate continuous function on a rectangular region in ${\mathbb{R}}^2.$ The parameters are chosen appropriately so that the corresponding fractal version preserves some properties inherent in the original function. We apply these results to invite the notion of bivariate fractal functions to the field of constrained approximation. Furthermore, we attempt to investigate the box dimension and Hausdorff dimension of the graph of the constructed bivariate fractal function.